Degenerate Elliptic Equations

0.1 The partial differential equation (1) (Au)(x) = L aa(x)(Dau)(x) = f(x) m lal9 is called elliptic on a set G, provided that the principal symbol a2m(X, ) = L aa(x) a lal=2m of the operator A is invertible on G X (~n 0); A is called elliptic on G, too. This definition works for systems of equations, for classical pseudo differential operators ("pdo), and for operators on a manifold n. Let us recall some facts concerning elliptic operators. 1 If n is closed, then for any s E ~ , is Fredholm and the following a priori estimate holds (2) 1 2 Introduction If m 0 and A : C=(O; C') -+ L (0; C') is formally self - adjoint 2 with respect to a smooth positive density, then the closure Ao of A is a self - adjoint operator with discrete spectrum and for the distribu tion functions of the positive and negative eigenvalues (counted with multiplicity) of Ao one has the following Weyl formula: as t -+ 00, (3) n 2m = t / II N±(1,a2m(x,e))dxde T·OO (on the right hand side, N±(t,a2m(x,e))are the distribution functions of the matrix a2m(X,e) : C' -+ CU).

Inhalt
0 Introduction.- 1 General Calculus of Pseudodifferential Operators.- 2 Model Classes of Degenerate Elliptic Differential Operators.- 3 General Classes of Degenerate Elliptic Differential Operators.- 4 Degenerate Elliptic Operators in Non Power Like Degeneration Case.- 5 Lp Theory for Degenerate Elliptic Operators.- 6 Coersiveness of Degenerate Quadratic Forms.- 7 Some Classes of Hypoelliptic Pseudodifferential Operators on Closed Manifold.- 8 Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary.- 9 General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equations.- 10 Spectral Asymptotics of Degenerate Elliptic Operators.- 11 Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics.- A Brief Review of the Bibligraphy.- Index of Notation.